An eigenvalue problem for derogatory matrices

摘要

A matrix A is called derogatory if there is more than one Jordan submatrix associated with an eigenvalue λ. In this paper, we are concerned with the eigenvalue problem of this type of matrices.The singularities of the resolvent of A:R(z)=(A-zI)-1 are exactly the eigenvalues of A. Let us consider the Laurent series of R expanded at λ and denote its coefficients ck(-∞⩽k⩽∞). ≔D≔c-2 is the nilpotent operator, that is, there exists the order l of λ such that ≔Dl≔c-l-1=0(l⩾1). Additionally, for an arbitrary vector z, Dl-1z is an eigenvector of λ. Then λ is computed from the corresponding eigenvector Dl-1z. In order to estimate the integral representation of Dkz, we apply the trapezoidal rule on the circle enclosing λ but excluding other eigenvalues of A.It is our result that, so far as related linear equations are solved with necessary precision, the eigenvalues of derogatory matrices can be computed numerically as exactly as we want and so are corresponding (generalized) eigenvectors, too.